338 XV NORMED ALGEBRAS AND SPECTRAL THEORY (15.4.9) Let G be a separable, metrizable, locally compact group. Then the mapping fju-+ (pi) = (ft)* (denoted by p) is an involution on the Banach algebra M£(G) (15.1.7); this follows from (14.7.3.1). Also we have ||ju|| = ||/Z|| = \\fi\\ (13.20.1), hence M£(G) is a Banach algebra with involution. It can be shown that it is not necessarily a star algebra. (15.4.10) It is clear that every self-adjoint subalgebra (resp. self-adjoint closed subalgebra) of a normed algebra with involution (resp. of a star algebra) is a normed algebra with involution (resp. a star algebra). (1 5.4.1 1 ) In an algebra A with involution, an element x is said to be self- adjoint (or hermitiari) if x* = x (the terminology is inspired by the example (15.4.7)). For each x e A, the elements x± = %(x + x*) and x2 = (x — x*)f2i are self-adjoint; so are xx* and x*x, and in general these two products are distinct. The element x is said to be normal if xx* = x*x, or equivalently if the self-adjoint elements jq and x2 commute. If A has a unit element e, then x e A is said to be unitary if xx* = x*x = e, that is if x is invertible and x'1 = x*. The unitary elements of A form a multiplicative group; for if x is unitary, then so is x"1 because (15.4.3) (x"1)* = (x*)"1 = x = (x"1)"1, and if x, y are unitary then so is xy9 because (xy)* = y*x* = y~"1x~1 = (xy)"1. (15.4.12) Let A be a Banach algebra with involution, having a unit ele- ment e T6 0. (i) For each xeA, Sp(x*) is the image of Sp(x) under the map- ping^ I (ii) If A is a star algebra, then for each self-adjoint element x e A we have Sp(x) c: R, and for each unitary element y e A we have Sp(y) c U. (i) If x — & is invertible, then so is (x - (e)* = x* — Z,e ((15.4.1) and (15.4.2)). Hence the result. (ii) Suppose that a 4- i(l e Sp(x), where a and ft are real. Then, for each real number A, the complex number a + /(/? + X) belongs to Sp(x + ite) and therefore, by (15.2.4), a2 + OS + I)2 g ||* + tie\\ 2 = \\x*x + A2£?|| g ||jt**|| + A2 = ||x||2 + A2, so that Since this is true for all real numbers A, it follows that /? = 0. On the other hand, if y is unitary, we have ||^||2 = ||>>*y|| = \\e\\ = 1, hence ||j|| = 1; similarly By"1!! = 1, and therefore Sp(y) c U (15.2.6).0 such that